"The Senate of the United States shall be composed of two Senators from each State, chosen by the Legislature thereof, for six Years; and each Senator shall have one Vote."
The purpose of the Senate was to ensure the equality of the states to each other and to the federal government. This was intended to counter the House where represenation was based on population size which allows some states to dictate terms to smaller states. The whole purpose of the Senate was to check the power of the House, including the House's power to impeach. In point of fact, the Senate has rejected all impeachments put forth by the House because nearly every impeachment was politically motivated.
In addition the Senate must approve all laws passed by the House of Representatives.
The Senate also serves as a check on the President because it has to approve all treaties, laws, and even appointments to the executive branch.
" Immediately after they shall be assembled in Consequence of the first Election, they shall be divided as equally as may be into three Classes. The Seats of the Senators of the first Class shall be vacated at the Expiration of the second Year, of the second Class at the Expiration of the fourth Year, and of the third Class at the Expiration of the sixth Year, so that one third may be chosen every second Year; and if Vacancies happen by Resignation, or otherwise, during the Recess of the Legislature of any State, the Executive thereof may make temporary Appointments until the next Meeting of the Legislature, which shall then fill such Vacancies."
We can break up this clause into two parts.
In the first part of the clause, the clause requires the Senate to be divided up into 3 electoral classs with a different class of Senate seats being up for election every 2 years. Every two years there is a Senate election happening somewhere in the USA.
The second half of the clause mandates how Senate vacancies are to be filled in the event of death or resignations. This clause empowers the governors of the states to temporarily make appointments to fill the seat until the legislature can meet but the ultimate authority to fill vacant Senate seats with unexpired terms is with the state legislature, not the governor and not the electorate. It is unconstitutional for the Governor of California to appoint someone to the remaining term of a Senator who has resigned and the US Constitution does not allow the legislature to allocate this power to the governor. Hence the governor's appointment of Kamala Harris' replacement, Alex Padilla, in California was unconstitutional and his occupation of that seat remains in direct violation of the US Constitution because the state legislature has not approved it.
"No Person shall be a Senator who shall not have attained to the Age of thirty Years, and been nine Years a Citizen of the United States, and who shall not, when elected, be an Inhabitant of that State for which he shall be chosen"
To serve in the US Senate you must at least 30 years old, have been a US citizen and a citizen of your state for at least 9 years, and live in the state you are seeking to represent.
"The Vice President of the United States shall be President of the Senate, but shall have no Vote, unless they be equally divided."
The primary job of the Vice President is to run the Senate, not to go globe trotting or hold rallies for the President. The failure of the Vice President, current and past, to carry out this constitutional obligation is unfortunate and remains a serious violation of the US Constitution.
"The Senate shall chuse their other Officers, and also a President pro tempore, in the Absence of the Vice President, or when he shall exercise the Office of President of the United States."
The name of the Senate leader in the absence of the Vice President is President Pro Tempore which means Temporary President. The title of the leader of the Senate is "Senate President" nor is it "President of the Senate". The official name of the title emphasizes the temporary nature of the post. The fact that certain people have been appointed on a permanent basis to be Senate President may actually be unconstitutional. However it should be noted this was changed by the 17th amendment passed by a later generation of American citizens. That amendment requires special elections to fill Se
"The Senate shall have the sole Power to try all Impeachments. When sitting for that Purpose, they shall be on Oath or Affirmation. When the President of the United States is tried, the Chief Justice shall preside: And no Person shall be convicted without the Concurrence of two thirds of the Members present."
The US Senate serves as a check on the House' impeachment powers which the House has too often abused historically. They abused in the Johnson impeachment, the abused in the Clinton impeachment and it was most recently abused in the Trump impeachments. When a sitting president is being impeached, the Constitution requires the Chief Justice of the US Supreme Court must preside. And at least 67% of Senators must vote to convict. In addition, even in the case of conviction the Senate alone decides the nature of the punishment. The House plays no role in deciding punishment. In fact the US Senate has rejected all impeachments proposed by the US House of Representatives because the impeachments were almost always carried out by political extremists or had such extremists on the prosecutorial team as the impeachment against Trump did.
"Judgment in Cases of Impeachment shall not extend further than to removal from Office, and disqualification to hold and enjoy any Office of honor, Trust or Profit under the United States: but the Party convicted shall nevertheless be liable and subject to Indictment, Trial, Judgment and Punishment, according to Law."
This clause limits the punishment for impeachment. And the punishment is actually reversible by a future Senate.
As Section 3 makes quite clear, the US Senate was to serve as a check on the power of the population majority based House because legislatures that based exclusively on population size are more likely to abuse their power than those that are based on equal representation between states.